It's detailed, but not overly complex.

Home is where the job starts. All tasks for that job are in home. Home is 
the dispatched address space. A task starts with H=P=S.
Any suitably authorized task (or SRB for that matter) may issue PC's, and 
the target of the PC may issue PC etc.
PC's may change P, do set S. But home is home.

The dispatched address space ASCB address is in PSAAOLD.
The dispatched task TCB address is in PSATOLD (PSATOLD = 0 for a 
dispatched SRB).
If I wanted the STOKEN of the home address space I would use
PSAAOLD -> ASCBASSB -> ASSBSTKN.
If I wanted the ASID of the home address space I would use
PSAAOLD -> ASCBASID.

In AR mode, as has been mentioned, there are 3 special ALETs. 2 are 
hardware-defined (0,1). The 3rd (2) is software-defined.
ALET=2 can be used to access storage in the home space when in AR mode 
(aside from timer DIE which is not supposed to rely on the dispatched 
space in any way).

For all I know, the snippet of code that showed use of LAM (et al) wasn't 
even in AR mode. 

I don't know anything about FASTAUTH's ACEEALET parameter, but it is 
certainly true that an ALET of 2 will access the home address space, and 
the home address space is where the TCB and ASXB and ACEE of the 
dispatched task / dispatched address space are.

As to why "ESAR" doesn't do what you want: it works as it is architected 
to work.  It gives the answer to "what is the secondary ASN". It just 
doesn't give an answer to "in which ASN is the TCB" in this case because 
(apparently in this case) H<>S.

Peter Relson
z/OS Core Technology Design

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